Owing to the development of electronic products, such as smart phones, tablet computers, and motion sensing gaming machines, and integration of electronic products, wireless communication, and broadband network, micro-electro-mechanical-system (MEMS) inertial sensors, e.g. accelerometer, gyroscope, and oscillator, became extensively used in these electronic products and the demand for MEMS inertial sensors has increased significantly in these years. The environments where the MEMS inertial sensors are used are subject to significant temperature change. The current trend is to develop an MEMS inertial sensor adaptable to different environment temperatures.
FIG. 1A is a simplified schematic view of a vibrating unit of an accelerometer. FIG. 1B is a simplified schematic view of a vibrating unit of an oscillator. Referring to FIG. 1A and FIG. 1B, these MEMS inertial sensor may be simplified as a vibrating unit composed of a flexure and a mass.
The resonance frequency of the vibrating unit can be obtained by the following equation:
  f  =            1                        2          ⁢                                          ⁢          π                      ⁢                  k        m            wherein f represents the resonance frequency of the vibrating unit, k represents the stiffness of the stiffness element, and m represents the mass of the mass. That is, the resonance frequency of the vibrating unit is determined by two important factors, i.e. the stiffness of the stiffness element and the mass of the mass.
The definition of the stiffness element is specified below. Among the elements of the vibrating unit, one can be defined as the stiffness element if the stiffness thereof is a critical factor for determining the resonance frequency of the vibrating unit. More specifically, in the aforementioned equation for calculating the resonance frequency of the vibrating unit, the stiffness parameter (k) is determined by the stiffness element.
Thus, in the accelerometer of FIG. 1A, the vibrating unit 10 is composed of a flexure 12 (spring) and a mass 14. The resonance frequency of the vibrating unit 10 is mainly determined by the stiffness of the flexure 12 and the mass of the mass 14. Therefore, in the accelerometer of FIG. 1A, the flexure 12 can be defined as the stiffness element.
In the oscillator of FIG. 1B, the vibrating unit 10a is a beam-shaped mass 12a and the resonance frequency of the vibrating unit 10a is mainly determined by the stiffness of the beam-shaped mass 12a and the mass of the beam-shaped mass 12a. Therefore, in the oscillator of FIG. 1B, the beam-shaped mass 12a can be defined as the stiffness element.
It can be known from the above that the stiffness of the stiffness element has a significant effect on the resonance frequency of the vibrating unit. The material properties, e.g. Young's modulus, of the stiffness element of the conventional MEMS inertial sensor are subject to change with the temperature, which may change the stiffness of the stiffness element and the resonance frequency of the vibrating unit, and consequently cause the obtained sensing value to be inaccurate. A variation of the Young's modulus due to temperature change may be represented by a temperature coefficient of Young's modulus (TCE). More specifically, the temperature coefficient of Young's modulus (TCE) may be defined as: the variation of the Young's modulus in a temperature unit.
FIG. 2 is a diagram showing the variation of a resonance frequency of a vibrating unit under different temperatures. When the resonance frequency of the vibrating unit changes with the change of temperature, the degree of change of the resonance frequency may be represented by a temperature coefficient of frequency (TCf), as shown in FIG. 2. Likewise, the temperature coefficient of frequency (TCf) may be specifically defined as: the variation of the resonance frequency in a temperature unit. When the resonance frequency of the vibrating unit does not change with the temperature, the temperature coefficient of frequency (TCf) of the vibrating unit is 0, which is called zero TCf.
FIG. 3 is a simplified schematic view of a micro-electro-mechanical-system (MEMS) resonator. Referring to FIG. 3, a stiffness element 22 of the MEMS resonator 20 is a beam-shaped mass. The stiffness element 22 is connected to an anchor 26 by a flexure 24 and driven by a driving electrode 28a to generate a periodical oscillation. When the beam-shaped mass reaches the state of resonance, the beam-shaped mass has the greatest amplitude of oscillation and the sensing electrode 28b senses the biggest voltage change, and the MEMS resonator 20 outputs the frequency of the measured voltage change as a clock signal. At the same time, the material properties, e.g. Young's modulus, of the stiffness element 22 change with the temperature, which changes the stiffness and the resonance frequency of the stiffness element 22 (beam-shaped mass). Consequently, the clock signal also changes with the temperature.
FIG. 4 is a simplified schematic view of a micro-electro-mechanical-system (MEMS) accelerometer. Referring to FIG. 4, when an MEMS accelerometer 30 is accelerated in the Y axis direction, a mass 34 that is connected to the anchor 26 by a stiffness element 32 (e.g. spring) is translated in the Y axis direction, and a relative distance between a stationary electrode 36a and a movable electrode 36b is changed due to the displacement of the mass 34. To be more specific, the change of the relative distance between the stationary electrode 36a and the movable electrode 36b results in change of capacitance between the stationary electrode 36a and the movable electrode 36b. Accordingly, the MEMS accelerometer 30 senses the capacitance change to calculate the acceleration. At the same time, the material properties, e.g. Young's modulus, of the stiffness element 32 change with the temperature, which changes the stiffness of the stiffness element 32. Thus, under the environments of different temperatures, the same acceleration excites the mass 34 of the MEMS accelerometer 30 to translate with different displacements and causes the capacitance change to be different. As a result, the acceleration value measured by the MEMS accelerometer 30 is inaccurate.
FIG. 5 is a simplified schematic view of a micro-electro-mechanical-system (MEMS) gyroscope. Referring to FIG. 5, a stiffness element 42 (spring) of the MEMS gyroscope 40 connects a frame 44a with a mass 44b and the stiffness element 42 (spring) of the MEMS gyroscope 40 connects a frame 44a and an anchor. The frame 44a oscillates along the Y axis direction at the resonance frequency thereof to drive the mass 44b to oscillate along the Y axis direction. When an angular velocity is applied in the Z axis direction, the mass 44b is translated in the X axis. Thus, a relative distance between a stationary electrode 46a and a movable electrode 46b changes. To be more specific, the change of the relative distance between the stationary electrode 46a and the movable electrode 46b results in change of capacitance between the stationary electrode 46a and the movable electrode 46b. Accordingly, the MEMS gyroscope 40 senses the capacitance change to calculate the angular velocity. At the same time, the material properties, e.g. Young's modulus, of the stiffness element 42 change with the temperature, which changes the stiffness of the stiffness element 42 and causes the resonance frequency of the peripheral frame 44a to change. Therefore, under the environments of different temperatures, the mass 44b has different displacements in the X axis direction and causes the capacitance change to be different when the same angular velocity in the Z axis direction is applied on the MEMS gyroscope 40. As a result, the angular velocity value measured by the MEMS gyroscope 40 is inaccurate.
From aforesaid examples, an issue in the development of MEMS inertial sensors, the current trend is to design an MEMS resonator having constant resonance frequency, an MEMS gyroscope having constant resonance frequency, or an MEMS accelerometer having constant spring stiffness even under different temperatures.
FIG. 6 is a simplified schematic view of a passive temperature compensated micro-electro-mechanical-system (MEMS) oscillator. Referring to FIG. 6, which discloses a passive temperature compensated MEMS oscillator 50, wherein SiO2 56 is filled into the trench 54 of the Si mass 52, so as to form the oscillating element (mass 52) with materials of two different Young's moduli. The Young's moduli of the two materials change with the temperature, and the variation of one of the Young's moduli is positive while the variation of another one is negative. Therefore, the resonance frequency of the MEMS oscillator 50 does not change with the temperature.
FIG. 7 is a simplified schematic view of another temperature compensated micro-electro-mechanical-system (MEMS) oscillator. Referring to FIG. 7, which discloses a composite MEMS oscillator 60, wherein four exterior surfaces of the Si oscillating element 62 are covered by SiO2 64 (see the cross-sectional view at the lower right corner of FIG. 7), so as to form the oscillating element 62 with materials of different Young's moduli. The Young's moduli of the two materials change with the temperature. Therefore, the resonance frequency of the MEMS oscillator 60 does not change with the temperature.
Further to the above, a paper “Temperature-Insensitive Composite Micromechanical Resonators” discloses an equation, as below, for calculating the temperature coefficient of frequency (TCf) of a composite resonator provided with a beam-shaped mass:
                    TCf        =                                            r              ·                              TCf                1                                      +                          TCf              2                                            r            +            1                                              (                  Equation          ⁢                                          ⁢          1                )                                r        =                                                            m                1                            ·                              f                1                2                                                                    m                2                            ·                              f                2                2                                              =                                                    E                1                            ·                              I                1                                                                    E                2                            ·                              I                2                                                                        (                  Equation          ⁢                                          ⁢          2                )                                          TCf          1                =                                            TCE              1                        +                          α              1                                2                                    (                  Equation          ⁢                                          ⁢          3                )                                          TCf          2                =                                            TCE              2                        +                          α              2                                2                                    (                  Equation          ⁢                                          ⁢          4                )            where E1 represents the Young's modulus of the mass, E2 represents the Young's modulus of the covering material, I1 represents the area moment of inertia of the mass, I2 represents the area moment of inertia of the covering material, TCf represents the temperature coefficient of frequency of the composite resonator, TCf1 represents the temperature coefficient of frequency of the mass, TCf2 represents the temperature coefficient of frequency of the covering material, TCE1 represents the temperature coefficient of Young's modulus of the mass, TCE2 represents the temperature coefficient of Young's modulus of the covering material, α1 represents the thermal expansion coefficient of the mass, α2 represents the thermal expansion coefficient of the covering material, f1 represents the resonance frequency of the mass, f2 represents the resonance frequency of the covering material, m1 represents the mass of the mass, and m2 represents the mass of the covering material.
The paper also discloses that, in order to prevent the resonance frequency of the composite resonator from changing with the temperature, the temperature coefficient of frequency (TCf) of the composite resonator has to be zero TCf. Based on the deduction below:
                              TCf          =                                                                      r                  ·                                      TCf                    1                                                  +                                  TCf                  2                                                            r                +                1                                      =            0                          ⁢                                  ⁢                  r          =                      -                                                            TCf                  2                                                  TCf                  1                                            .                                                          (                  Equation          ⁢                                          ⁢          5                )            
It is known from the above that the composite vibrating unit in the common passive temperature compensated MEMS oscillator is formed by using a silicon material and the first material (e.g. SiO2). In the case where the variation direction of the Young's modulus of the first material is opposite to the variation direction of the Young's modulus of the silicon material when the temperature changes, and the area moment of inertia of the mass of the composite resonator and the area moment of inertia of the covering material satisfy Equation 2 and Equation 5, the stiffness of the stiffness element (beam-shaped mass) in the oscillating direction remains unchanged, and the resonance frequency of the beam-shaped mass does not change with the temperature. As a result, the clock signal generated by the oscillator does not change with temperature.